Say I have a hyperelliptic curve without any automorphism beyond the hyperelliptic involution.
Is it possible for its symmetric square to obtain new automorphisms beyond the one induced by the hyperelliptic involution?
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$\begingroup$ A general $g^2_4$ on a genus $2$ curve gives an unramified morphism from the curve to a plane quartic with a single ordinary double point. Plane quartics have an involution on their symmetric square, just by taking the residual of a degree $2$-divisor in the intersection of the quartice with the secant line. $\endgroup$– Jason StarrCommented Sep 7, 2023 at 12:26
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$\begingroup$ I see now that the involution in my comment (which is, indeed, different from the hyperelliptic involution) is only defined on the open complement of a singleton: it is defined for all points in the symmetric square except a single point. So this is a birational involution of the symmetric square, not a biregular involution. $\endgroup$– Jason StarrCommented Sep 7, 2023 at 21:48
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